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Chapter 1

A short review of AlgebraicGeometry

1.1 Affine Varieties

In algebraic geometry one studies the connections between algebraic varieties,which are sets of solutions of polynomial equations, and complex algebras.

An affine variety is a subset X ⊂ Cn that is defined by a finite set of polynomialequations.

X := {x ∈ Cn|f1(x) = 0, . . . , fk(x) = 0}

A morphism between two varieties X ∈ Cn an Y ∈ Cm is a map φ : X → Ysuch that there exist a polynomial map Φ : Cn → Cm such that φ = Φ|X . Sucha morphism is an isomorphism if φ is invertible and φ−1 is also a morphism.

We can consider C as a variety, so it makes sense to look at the morphisms froma variety X to the variety C, these maps are also called the regular functions onX. They are closed under pointwise addition and multiplication so they form acommutative C-algebra: C[X].

C[X] := {φ : X → C|ρ is a morphism of varieties}

This algebra can be described with generators and relations. To every varietyX ∈ Cn the set of polynomial functions that are zero on X form an ideal inC[x1, · · · , xn]. If we divide out this ideal we get the ring of polynomial functionson X.

C[X] := C[x1, . . . , xn]/(f |∀x ∈ X : f(x) = 0)

1

CHAPTER 1. A SHORT REVIEW OF ALGEBRAIC GEOMETRY

This algebra is finitely generated by the xi and it also has no nilpotent elementsbecause f(x)n = 0⇒ f(x) = 0.

A morphism between varieties, φ : X → Y , will also give an algebra morphismbetween the corresponding rings but the arrow will go in the opposite direction:

φ∗ : C[Y ]→ C[X] : g 7→ g ◦ φ.

On the other hand if R is a finitely generated commutative C-algebra withoutnilpotent elements, by definition we will call this an affine algebra. Every affinealgebra can be written as a quotient of a polynomial ring C[x1, . . . , xn] with anideal i. Because polynomial rings are Noetherian, i is finitely generated by f.i.f1, . . . , fk. Therefore we can associate to R the variety V (R) in Cn defined bythe fi.

Although this variety depends on the choice of generators of R and i, there isa more intrinsic description of V (R). Indeed the points of V (R) are in one toone correspondence with the algebra morphisms from R to the algebra C (orequivalently the maximal ideals of R.)

V (R) := {ρ : R→ C|ρ is an algebra morphism}

A morphism between algebras, φ : R → S, will also give an algebra morphismbetween the corresponding rings but the arrow will go in the opposite direction:

φ∗ : V (S)→ V (R) : ρ 7→ ρ ◦ φ.

The main theorem of algebraic geometry now states that the operations V (−)and C[−] are each other’s inverses:

Theorem 1.1. The category of commutative affine algebras and the category ofaffine varieties are anti-equivalent. So working with affine varieties is actuallythe same as working with affine algebras but all maps are reversed. The anti-equivalence is given by the contravariant functors V (−) and C[−], so

C[V (R)] ∼= R and V (C[X]) ∼= X

This theorem enables us to translate every geometrical statement about affinevarieties into an algebraic about affine algebras and vice versa.

Memo 1.2. Affine algebraic geometry is affine commutative algebra with thearrows reversed.

2


1.2 Projective Varieties

Given a set of variables x1, . . . , xn with degree 0, a set of variables y0, . . . , ym ofdegree 1, and a set of homogeneous polynomials f1, . . . , fk. We define a subset ofCn × Pm where Pm is the projective m-space.

V := {((x1, . . . , xn), (y0 : · · · : ym)) |fi = 0}

This is well defined because the polynomials are homogeneous. To this we canassociate the ring

R = C[x1, . . . , xn, y0, . . . , ym]/(f1, . . . , fk).

This ring is graded by giving the xi degree 0 and the yi degree 1.

Y = Proj R := {maximal graded ideals in R that do not contain the ideal (y0, . . . , yn)}

The ring R contains a subring R0 of all degree 0 elements (the ring generated bythe xi) and there is a map from Y to X := V (R0):

π : Y 7→ V (R0) : ((x1, . . . , xn), (y0 : · · · : ym))→ (x1, . . . , xn).

We call Y a variety that is projective over X. If R0∼= C then V (R0) is a point

and we call Y projective. Such a variety is the union of affine varieties:

Yi := {((x1, . . . , xn), (y0 : · · · : ym)) ∈ Y |xi = 1} = V (R/(yi − 1)).

A morphism between two such varieties Y and Y ′ is a map φ : Y → Y ′ such thatφij : φ−1(Y ′j ) ∩ Yj → Y ′j : x 7→ φ(x) is a morphism between affine varieties forall i, j. Such a morphism is an isomorphism if φ is invertible and φ−1 is also amorphism.

An example of an isomorphism of two projective varieties is the following. LetX = P1 and Y := {(x : y : z)|xy − z2 = 0} and define

φ : X → Y : (x : y) 7→ (x2 : y2 : xy).

However unlike in the affine case there are non-isomorphic rings that give isomor-phic projective varieties: in the example above Proj C[x, y] = X and Proj C[x, y, z]/(xy−z2) = Y .

The advantage of using projective varieties is that as topological spaces they arecompact, while the only affine varieties that are compact are points. Using thesevarieties makes some geometry usual simpler (affine, there is a differents betweenthe hyperbola and parabola but there projective versions are isomorphic).

Apart from projective and affine varieties there are lots of other varieties, but thegeneral definition of a variety is quite complicated, so we will not introduce it.Intuitively a variety is a topological set which is the union of overlapping affinevarieties.

3


1.3 Smoothness

If V ⊂ Cn is an affine variety defined by f1, . . . , fk then we define the tangentspace in p ∈ V as the vector space

TpV := {(y1, . . . , yn)|(∂fi∂xj

)p

yj = 0}

Again this definition depends on the choice of the f ′s but one can show that adifferent choice of generators gives an isomorphic vector space.

The dimension of a variety V is the minimal dimension of all its tangents spaces.

dimV := minp∈V

dimTpV

A point p is called a smooth point if dimV = dimTpV and otherwise it is calledsingular. A vbariety is called smooth if all its points are smooth and singularotherwise. The subset of all singular points of a variety is called the singularlocus.

Smooth varieties are in many ways the nicest and easiest varieties and singularvarieties are a lot more difficult to study. Therefore one wants to find a methodto turn a singular variety V into a closely related smooth variety V . This processis called resolving the singularities. More precisely one wants to construct asurjective morphism V → V that is almost everywhere bijective (i.e. on an opendense part) such that all the fibers are compact and V is smooth.

4

Chapter 2

Resolving singularities

2.1 Resolving singular curves

The first example we do is a singular elliptic curve. LetR be the ring C[X, Y ]/(Y 2−X3 +X2). This curve has a singular point in (0, 0).

The problem seems that there are two tangent directions in (0, 0), and the curveintersects itself. To get rid of this singularity one should split to point (0, 0) intwo. This can be done by adding the function Y/X (the slope of the tangent in0) to the ring

R := C[X, Y,Y

X]/(Y 2 −X3 −X2) = C[X, Y, Z]/(Y 2 −X3 +X2, ZX − Y )

What is special about the element Z? it is an element of the quotient field Q(R) :={ab|a, b ∈ R} and it is integral over R i.e. it satisfies a monic polynomial with

coefficients in R:Z2 − (X − 1) = 0.

In fact one can check that the ring R is the integral closure of R: it is the subringof Q(R) consisting of all elements that are integral over R. A ring that equals itsintegral closure is called integrally closed or normal.

Theorem 2.1 (normal curves are smooth). If R is a normal ring and V (R) hasdimension 1 then V (R) is smooth.

Proof. See Janos Kollar, Lectures on resolution of singularities. Theorem 1.30

If R is the coordinate ring of a singular curve then we can look at the embeddingof R in its integral closure. The embedding ι : R→ R gives us a map π : V (R)→

5

CHAPTER 2. RESOLVING SINGULARITIES

V (R). One can prove that this map is surjective and a bijection on the smoothpoints of V (R).

Memo 2.2. Resolving a singular curve can be done by going to the integralclosure.

2.2 Resolving surface singularities

In dimension 2 going to the integral closure is not sufficient because there aresingular varieties for which the ring is integrally closed. The standard exampleof this is C[x, y, z]/(xy − z2).

If we to generalize the constructions of dimension 1, we would like to add onepoint for each tangent line through a singularity. This procedure cannot be doneby affine geometry alone because the space of lines through a point is a projectivevariety.

An interesting way of constructing resolutions is by using blow-ups. Suppose Vis an affine variety and n C C[V ] is an ideal corresponding to the closed subsetX. The blow-up of X is then defined as

V = Proj C[V ]⊕ nt⊕ n2t2 ⊕ · · · .

The standard projection π : V → V is at least a partial resolution because ifp ∈ V \X and y0t, . . . ymt are the generators of nt is there must be at least oneyi that is not zero on p, so the preimage of p will only contain the point

(x1(p), . . . , xn(p), y0(p), . . . , ym(p))

We will now do some examples. The ring C[X, Y, Z]/(XY − Z2) has a uniquesingularity in the point (0, 0, 0) because

(∂X , ∂Y , ∂Z)r = (Y,X, 2Z) = 0⇔ (X, Y, Z) = (0, 0, 0).

The blow-up is (using the convention x = Xt, y = Y t, z = Zt)

Proj C[X, Y, Z]/(r)⊕ (X, Y, Z)t⊕ (X, Y, Z)2t2 ⊕ · · ·

=C[X, Y, Z, x, y, z]

(XY − Z2, Xy − xY, xZ −Xz, Y z − yZ,Xy − Zz, xy − z2)

= {(X, Y, Z,X, Y, Z) ∈ C3 \ {0} × P2|XY − Z2} ∪ {(0, 0, 0, x, y, z) ∈ P2|xy − z2 = 0}

where the last bit is the exceptional fiber, it is a conic and hence as a variety it isisomorphic to P1. We can cover the blow-up variety by two parts correspondingto

6


x 6= 0 we can put x = 1 and then the ring becomes

R/(x−1) =C[X, Y, Z, y, z]

(XY − Z2, Xy − Y, Z −Xz, Y z − yZ,Xy − Zz, y − z2)= C[X, z]

which is smooth.

y 6= 0 we can put y = 1 and then the ring C[Y, z].

z 6= 0 is not necessary because it implies that both x, y 6= 0.

The ring An = C[X, Y, Z]/(XY −Zn), n ≥ 3 has a unique singularity in the point(0, 0, 0) because

(∂X , ∂Y , ∂Z)r = (Y,X, 3Z2) = 0⇔ (X, Y, Z) = (0, 0, 0).

The blow-up is

C[X, Y, Z, x, y, z]

(XY − Zn, Xy − xY, . . . , Xy − Zn−1z, xy − Zn−2z2)

= {(X, Y, Z,X, Y, Z) ∈ C3 \ {0} × P2|XY − Z2} ∪ {(0, 0, 0, x, y, z) ∈ P2|xy = 0}

where the last bit is the exceptional fiber, it is a union of 2 projective lines thatintersect in the point (0, 0, 0, 0, 0, 1).

We can cover the blow-up variety by three parts corresponding to

x 6= 0 gives C[X, z] which is smooth.

y 6= 0 gives C[Y, z] which is smooth.

z 6= 0 gives C[Z,x,y](xy−Zn−2)

which has a singularity if n > 3, but this singularity is’smaller’ so we can blow it up again.

Diagramatically we get the following

�

An

PWQVRUSTPWQVRUST�

An−2

PWQVRUSTPWQVRUSTPWQVRUSTPWQVRUST• •�

An−4

PWQVRUSTPWQV RUSTPWQVRUST• •· · ·

n− 1× P1

Memo 2.3. The singularities C[X, Y, Z]/(XY −Zn) can be resolved by blowingup several times.

7


Now have a look at Dn := C[X, Y, Z]/(Xn+1 +XY 2 + Z2)

If n = 2 The first blow-up has exceptional fiber z = 0 because the relationbecomes Xx2 + Xy2 + z2 = 0. If we look at the chart for y 6= 0 we get therelation

x3Y + xY + z2

which has three singularities, for x = 0,±i if we blow these 3 up, we get 3exceptional fibers of the form ξy+ ζ2 with ζ = zt, ξ = t(x+ 0,±i) (depending onthe point blown up). One can check easily that there are no further singularities.

If n > 2 then the exceptional fiber is z = 0. There are two singularities in theblow-up, the one corresponding to (1, 0, 0) which is of the type Dn−2 and the onein (0, 1, 0) which has local equation xn+1Y n−1 +xY +z2. The blow-up of this lastsingularity has as exceptional fiber a conic (look at the degree 2-part) and thereare no further singularities.

The diagram looks as

N

Dn

PWQVRUSTN �

Dn−2

PWQVRUSTPWQVRUSTPWQVRUST• N �

Dn−4

PWQVRUSTPWQV RUSTPWQVRUST• •· · · N

D3

PWQVRUSTPWQV RUSTPWQVRUSTPWQVRUST• •· · · NN

NC

PWQVRUSTPWQV RUSTPWQVRUSTPWQVRUSTPWQVRUSTPWQVRUST

PWQVRUST• •· · · • •

•

•

PWQVRUSTPWQV RUSTPWQVRUST• •· · · N

D2

PWQVRUSTPWQV RUSTPWQVRUSTPWQVRUST• •· · · •N

NC

These two examples indicate that in dimension 2 the way to resolve singularities,is by blowing up singular points. This is indeed the case in general:

Memo 2.4. Resolving a singular surface can be done by consecutively going tothe intgral closure and blowing up singular points.

In general this process can be quite cumbersome and tedious, so we want tosearch for a new way of constructing the resolution. To do this we need to havea look at group actions.

8

Chapter 3

Group actions and the affinequotient

3.1 Quotients and rings of invariants

Suppose we have a finite group G and let V be a finite dimensional complex vectorspace with a linear G-action (i.e. for every g ∈ G the map v 7→ g · v is linear.

V can also be considered as the variety Cn. For every point v ∈ V we can definethe orbit G · v := {g · v|g ∈ G}. Orbits never intersect so we can partition V intoits orbits. We will denote the set of all orbits by V/G.

A natural question one can ask if whether this set can also be given the structureof an affine variety. In the case of finite groups it will be possible, but for generalgroups there will be extra complications.

We can take a closer look at the problem by looking at the algebraic side of thestory. The ring of polynomial functions over V is R = C[V ] ∼= C[X1, . . . , Xk] is agraded polynomial ring if we give the Xi degree 1.

On R we have an action of G:

G× C[V ]→ C[V ] : (g, f) 7→ g · f := f ◦ ρV (g−1).

This action is linear and compatible with the algebra structure: g · f1f2 = (g ·f1)(g · f2).

The set of elements of R that are invariant under the group form a sub ring ofR, which we call the ring of invariants and we denote it by

RG := {f |g · f = f}

9

CHAPTER 3. GROUP ACTIONS AND THE AFFINE QUOTIENT

The G-action maps homogeneous elements of to homogeneous elements with thesame degree and therefore the ring of invariants is alsoo a graded ring.

We are now ready to state the main theorem:

Theorem 3.1. If G is a finite group with a linear group action V then the ringof invariants S = C[V ]G is finitely generated.

Proof. To prove that RG is finitely generated we first prove that this ring isnoetherian. Suppose that

a1 ⊂ a2 ⊂ a3 ⊂ · · ·is an ascending chain of ideals in RG. Multiplying with R we obtain a chain ofideals in R:

a1R ⊂ a2R ⊂ a3R ⊂ · · · .This chain is stationary because R is a polynomial ring and hence noetherian.

Now construct the map

% : R 7→ RG : f 7→ 1

|G|∑g∈G

g · f.

This map is called the reynolds operator and has the property that π(f) = f ifand only if f ∈ RG and π(f1f2) = f1π(f2) if f1 ∈ RG. Therefore π(aiR) = ai andhence the chain a1 ⊂ a2 ⊂ a3 ⊂ · · · is also stationary.

Now let S+ = denote the ideal of RG generated by all homogeneous elements ofnonzero degree. Because RG is Noetherian, S+ is generated by a finite numberof homogeneous elements: S+ = f1R

G +. . . +frRG.Wewillshowthatthesefi also

generate S as a ring.

Now RG = C + S+ so S+ = Cf1 + · · ·+ Cfr + S2+, S2

+ =∑

i,j Cfifj + S3+ and by

inductionSt+ =

∑i1...it

Cfi1 · · · fit + St+1+ .

So C[f1, . . . , fr] is a graded subalgebra of RG and RG = C+S+ = C[f1, . . . , fr]+St+

for every t. If we look at the degree d-part of this equation we see that

RGd = C[f1, . . . , fr]d + (St+)d.

Because St+ only contains elements of degree at least t, (St+)d = 0 if t > d. Asthe equation holds for every t we can conclude that

RGd = C[f1, . . . , fr]d and thus RG = C[f1, . . . , fr]

10


Now because RG is finitely generated and does not have nilpotent elements, itcorresponds to a variety V (RG) and the embedding RG ⊂ R gives a

V (R)→ V (RG) : m 7→ m ∩RG

This map is a surjection because if s is a maximal ideal in RG then sR is not equalto R because then π(sR) = sπ(R) = s 6= RG. Therefore sR will be contained ina maximal ideal mCR (there may be more) so π(m) = s.

Furthermore points of V (R) in the same orbit are mapped to the same point inV (RG) because f(g · p) = (g · f)(p) = f(p) if f ∈ RG. The reverse implicationis also true. Remember that for a finite number of points p1, . . . , pk and a set ofcomplex numbers a1, . . . ak we can always find a polynomial function f ∈ R suchthat f(pi) = ai. So if p and q have disjoint orbits we can chose the values suchthat ∑

g∈G

f(gp) 6=∑g∈G

f(gq).

therefore the function %(f) ∈ RG will be different on the orbits of p and q.

Memo 3.2. We can construct a quotient of a finite group by taking the varietycorresponding to the ring of invariants.

3.2 Infinite groups

For infinite groups the situation is a bit more complicated. In full generality thering of invariants is not finitely generated, but for a large class of groups it still is.Such groups are call reductive groups. Examples of infinite reductive groups arcompact groups, general linear groups, orthogonal groups and symplective groupsand finite cartesian products of them. There are however nonreductive groups ofwhich C,+ is the most important.

For reductive groups we still get a surjection from V (R) to V (RG) but morethan one orbit can be mapped to the same point. Because the surjection is acontinuous map π−1(x) must be a closed subset, so if there exists an orbit O thatis not closed and w = π(O) we know that π−1(w) must contain points outsideO (Note that if G is finite then this problem does not occur because all orbitscontain only a finite number of points and are hence closed). However it is stilltrue that different closed orbits are mapped to different points.

We can summarize all this in a theorem

Theorem 3.3. If V is a finite dimensional representation of a reductive group,then there exists a unique variety V//G = V (C[V ]G) such that

11


1. The points are in one-to-one correspondence with the closed orbits in V .

2. The projection V → V//G is a categorical quotient.

3. If G is finite then as a set V//G = V/G.

3.3 Examples

In this section we will determine generators and relations for the rings of invari-ants of some group actions. We will do two examples consisting where the groupis a group of 2× 2-matrices which act on V = C2.

I is generated by g =

(e

2πin 0

0 e−2πin

)

II is generated by g =

(eπin 0

0 e−πin

)and s =

(0 ii 0

)

In order to find the generator and relations of the rings of invariants, we will usethe Reynolds operator

%(f) =1

|G|∑g∈G

g · f.

This map is a projection %2 = % and it is the identity operation on C[V ]G. So toget a basis for the ring of invariants we can look at the set of images of all themonomials in C[V ].

%X iY j.

We will now consider the different types

I If g is the generator of the cyclic group then g ·X = ζX, g · Y = ζ−1Y withζ = e2π/n. Therefore

%X iY j =n∑k=1

gk ·X iY j

=n∑k=1

ζk(i−j)X iY j

=

{0 i 6= j mod n

nX iY j i = j mod n.

12


From this one can deduce that all invariants are generated by ξ = Xn, η =Y n and ζ = XY . One can easily see that there is a relation between thethree invariants ξη − ζn.

II Because s2 = gn = −1 and gs = −sg, the elements of this group can bewritten as sigj with i = 0, 1 and j = 1, . . . , 2n.

Therefore

%X iY j =2∑

k=1

ngk ·X iY j + sgk ·X iY j

=n∑k=1

ζk(i−j)(X iY j + ii+jXjY i)

=

0 i 6= j mod 2n or

X iY j − Y jX i i = j mod 2n and i is odd

X iY j + Y jX i i = j mod 2n and i is even

.

From this one can deduce that all invariants are generated by ξ = X2Y 2, η =XY (X2n − Y 2n) and ζ = X2n + Y 2n. One can easily see that there is a relationbetween the three invariants ξn+1 − ξη2 + ζ2.

In both cases, the dimension of the quotient space must be two because the mapC2 → V/G has finite fibers. If the ideal would be generated by more than onegenerator C[ξ, η, ζ]/p, its corresponding variety would not be two-dimensional.

As we notice here, these singularities are precisely the ones we tried to blow upin the previous chapter. In the following chapters we will use this viewpoint fora new way to construct resolutions for the singularity.

13


14

Chapter 4

Smash Products and Quivers

4.1 Smash Products

As we know from the previous chapter, the action of G on V gives rise to an actionon the polynomial ring C[V ] ∼= C[X, Y ]. Construct the vector space C[V ]|G|. Wecan identify the standard basis elements with the elements in the group, such thatevery element of this space can be written uniquely as a sum of f(X, Y )g wheref(X, Y ) is a polynomial function and g is an element of G. We can now define aproduct on this vector space

fi(X, Y )gi × fj(X, Y )gj = (figi · fj)gigj,

in this expression the · denotes the action of G on C[V ]. One can easily check thatthis product is associative and by linearly extending it to the whole vector spaceone obtains an algebra: the smash product of C[V ] and G. In symbols we writeC[V ]#G. The center of this algebra can be easily determined: if z =

∑g fgg ∈ Z

then

∀f ∈ C[V ] : [z, f ] = fg(f − g · f)g = 0 and ∀h ∈ g : [z, h] = (h · fg − fg)gh

The first equation implies that fg = 0 if g 6= 1 and the second implies that f1

must be a G-invariant function so we can conclude that

Z(C[V ]#G) ∼= C[V ]G.

This algebra has a nice description using quivers.

15

CHAPTER 4. SMASH PRODUCTS AND QUIVERS

4.2 Quivers

A quiver Q = (Q0, Q1, h, t) consists of a set of vertices Q0, a set of arrows Q1

between those vertices and maps h, t : A → V which assign to each arrow itshead and tail vertex. We also denote this as

76540123h(a) 76540123t(a)aoo .

A sequence of arrows a1 . . . ap in a quiver Q is called a path of length p if t(ai) =h(ai+1), this path is called a cycle if t(ap) = h(a1). A path of length zero will bedefined as a vertex. A quiver is strongly connected if for every couple of vertices(v1, v2) there exists a path p such that s(p) = v1 and t(p) = v2.

If we take all the paths, including the one with zero length, as a basis we canform a complex vector space CQ. On this space we can put a noncommutativeproduct, by concatenating paths. By the concatenation of two paths a1 . . . ap andb1 . . . bq we mean

a1 . . . ap · b1 . . . bq :=

{a1 . . . apb1 . . . bq s(ap) = t(b1)

0 t(ap) 6= h(b1)

For a vertex v and a path p we define vp as p if p ends in v and zero else. On theother hand pv is p if this path starts in v and zero else.

The vector space CQ equipped with this product is called the path algebra. Theset of vertices Q0 = {v1, . . . , vk} forms a set of mutually orthogonal idempotentsfor this algebra. The subalgebra generated by these vertices is isomorphic toCQ0 = C⊕k and this is also the degree zero part if we give CQ a gradation usingthe length of the paths.

An algebra A is called a path algebra with relations A is isomorphic to thequotient of a path algebra by an ideal sitting inside CQ≥2, which is the spacespanned by all paths of length at least 2.

Theorem 4.1. If G is a finite abelian group with a linear action on V , then thesmash product is a path algebra of a quiver with relations.

Proof. Sketch. Let G be the set of group morphisms from G to C∗. for eachelement φ ∈ G we define

eφ :=1

G

∑g∈G

φ(g)g

One can prove that eφeψ = 0 if φ 6= ψ and e2φ = eφ. Moreover∑

φ∈G eφ = 1.

16


This means that the eφ play the role of the vertices.

Because G is abelian we can find a basis X1, . . . , Xk for V such that the Xi areeigenvectors for all of the g ∈ G. This means that every Xi there is a φi ∈ G suchthat g ·Xi = φi(g)Xi.

One can check that eφXi = Xieφiφ = eφXieφiφ and therefore eφXi can be seen asan arrow with head eφ and tail eφiφ.

This means that Xi =∑

φ eφXi splits as the sum of |G| arrows one for each eφ.

In a similar way we can split the relations XiXj − XjXi as a sum of paths oflength two in these arrows.

remark 4.2. If G is not abelian this is not true anymore. But something verysimilar is going on. The statement now becomes that the smash product A =C[V ]#G contains an idempotent e such that eAe is a path algebra with relationsand A = AeA (which implies that the idempotent does not map anything essentialto zero).

4.3 representations of quivers

A dimension vector of a quiver is a map α : Q0 → N, the size of a dimensionvector is defined as |α| :=

∑v∈Q0

αv. A couple (Q,α) consisting of a quiver anda dimension vector is called a quiver setting and for every vertex v ∈ Q0, αv isreferred to as the dimension of v. A setting is called sincere if none of the verticeshas dimension 0. For every vertex v ∈ Q0 we also define the dimension vector

εv : V → N : w 7→

{0 v 6= w,

1 v = w.

An α-dimensional complex representation W of Q assigns to each vertex v a linearspace Cαv and to each arrow a a matrix

Wa ∈ Matαh(a)×αt(a)(C)

The space of all α-dimensional representations is denoted by Rep(Q,α).

Rep(Q,α) :=⊕a∈A

Matαh(a)×αt(a)(C)

To the dimension vector α we can also assign a reductive group

GLα :=⊕v∈V

GLαv(C).

17


An element of this group, g, has a natural action on Rep(Q,α):

W := (Wa)a∈A, Wg := (gt(a)Wag

−1s(a))a∈A

The quotient of this action will be denoted as

iss(Q,α) := Rep(Q,α)//GLα.

A representationW is called simple if the only collections of subspaces (Vv)v∈V , Vv ⊆Cαv having the property

∀a ∈ A : WaVs(a) ⊂ Vt(a)

are the trivial ones (i.e. the collection of zero-dimensional subspaces and (Cαv)v∈V ).

The direct sum W ⊕W ′ of two representations W,W ′ has as dimension vectorthe sum of the two dimension vectors and as matrices (W ⊕W ′)a := Wa ⊕W ′

a.A representation equivalent to a direct sum of simple representations is calledsemisimple.

A very important theorem is:

Theorem 4.3. A representation of a quiver is semisimple if and only if its orbitis closed in Rep(Q,α).

Proof. omitted

If we have a path algebra with relations A we define Rep(A,α) as the subset ofRep(Q,α) which respect the relations. We will denote this variety as Rep(A,α).Because this is a closed subset of Rep(Q,α), the quotient Rep(A,α)//GLα can beseen as the image of Rep(A,α) under the quotient map Rep(Q,α) → iss(Q,α).Again it classifies the semisimple α-dimensional representations up to isomor-phism and hence we denote it by iss(A,α).

4.4 The representation space of an abelian smash

product

In this section we do some examples in the case of abelian smash products

• Let G be Z2 which acts on C2 by gX = −X and gY = −Y . The quiverlooks like

�� x1,x2

%-��y1,y2

em

18


with relations x1y2 − x2y1 and x1y2 − x2y1.

The representation space with dimension vector (1, 1) is

{(a1, a2, b1, b2) ∈ C4|a1b2 − a2b1}.

This space is threedimensional.

The invariant functions on Rep(A,α) are generated by X = a1b1, Y = a2b2and Z = a1b2 which is the same as a2b1. The relation between theseinvariants is XY − Z2 so the ring of invariants is indeed C[iss(A, )] =C[X, Y, Z]/(XY − Z2).

For each point (x, y, z) 6= (0, 0, 0) we can construct a simple representation

– if z 6= 0 then we take (1, y/z, x, z).

– for (x, y, z) = (1, 0, 0) we take (1, 0, 1, 0).

– for (x, y, z) = (0, 1, 0) we take (0, 1, 0, 1).

For the point (x, y, z) = (0, 0, 0) there are several orbits:

– {(0, 0, 0, 0)} which is a point

– For each (x : y) ∈ P1 we have an orbit {(ax, ay, 0, 0)|a ∈ C∗}.– For each (x : y) ∈ P1 we have an orbit {(0, 0, ax, ay)|a ∈ C∗}.

19


20

Chapter 5

Semi-invariants and Modulispaces

5.1 Semi-invariants

As we have seen in the previous chapter it is possible to get a resolution of anaffine variety by constructing the Proj of a graded ring of which the degree zeropart is the ring of regular functions of the original variety. The method we usedfor this was blow-ups. In invariant theory it is also possible to do a differentconstruction using semi-invariants.

If G is a reductive group then a multiplicative character of G is a group morphismθ : G→ C∗ : g 7→ gθ. We will write the action of θ exponentially because it willbe very handy later on. The characters of G form an additive group if we definegθ1+θ2 := gθ1gθ2 , we will also use the shorthand nθ = θ + · · ·+ θ.

If G acts on a variety V then a function f ∈ C[V ] is called a θ-semi-invariant if

∀g ∈ G : g · f = gθf.

The subspace of θ-semi-invariants will be denoted by C[V ]θ. This space does notform a ring, it is only a module over the ring of invariants C[V ]G.

We can construct an N-graded ring by taking the direct sum of all nθ-semi-invariants with n ∈ N:

SIθ[V ] =⊕n∈N

C[V ]nθ.

It is easy to extend the proof of theorem to show that SIθ[V ] is also finitelygenerated as an algebra over SIθ[V ]0 = C[V ]G.

21

CHAPTER 5. SEMI-INVARIANTS AND MODULI SPACES

A point p ∈ V is called θ-semi-stable if there is an f ∈ C[V ]nθ such that f(p) 6= 0.The set of θ-semi-stable points will be denoted by V ss

θ . Note that V ssθ itself is

not necessarily an affine variety but if f1, · · · , fk forms a set of homogeneousgenerators of SIθ[V ] over C[V ]G then V ss

θ can be covered with affine varietiescorresponding to the rings

Ri = C[V ][f−1i ] V (Ri) = {p ∈ V |fi(p) 6= 0}

These varieties and rings have G-actions on them coming from the G-action onV and one can take the categorical quotient of these varieties. Their ring are ofthe form

SIθ[V ][f−1i ]0

and hence one can cover Proj SIθ[V ] with these quotients varieties. Out of thisone can conclude

Theorem 5.1. The variety Proj SIθ[V ] classifies the closed orbits in V ssθ . If there

exists a θ-semi-invariant that is non-zero in a point of V then V ssθ is open and

dense in V and the image of the map V ssθ //G→ V//G is dense.

5.2 semi-stable representations of quivers

If Q is a quiver and α a dimension vector then we can look at the θ-semi-invariantsof the GLα-action on Rep(Q,α). We will denote this set by Repssθ (Q,α), thequotient of this set by the GLα-action we be denoted by Mss

θ (Q,α) and is calledthe moduli space of θ-semistable representations.

First of all we have to look at the multiplicative characters of GLα. For the generallinear group GLn the characters are given by powers of the determinant, so thegroup of characters is isomorphic to Z. As GLα consists of k = #Q0 componentseach one isomorphic to a general linear group, the group of characters will beisomorphic to Zk:

θ = (θ1, . . . , θk) : GLα → C∗ : (M1, . . . ,Mk) 7→ detM θ11 · · · detM θk

k .

In the case of invariants we had a nice description using traces of cycles, for semi-invariants we can do a similar thing. A way to construct a θ-semi invariant is thefollowing: let i1, . . . , is be the vertices for which θi` is positive, while j1, . . . , jt bethe ones with a negative θj` . Now chose for each i and j |θiθj| elements in jCQi

22


and put all these in a∑

j θj ×∑

i θi-matrix D over CQ.

D :=

j1←i1 ... j1←i1 j1←is ... j1←is... |θj1θi1 |×

... ···... |θj1θis |×

...j1←i1 ... j1←i1 j1←is ... j1←is

......

...jt←i1 ... jt←i1 jt←is ... jt←is

... |θjtθi1 |×... ···

... |θjtθis |×...

jt←i1 ... jt←i1 jt←is ... jt←is

Now if W ∈ Rep(Q,α) then we can substitute each entry in D to its correspondingmatrix-value in W . In this way we obtain a block matrix DW with dimensions∑

i αi|θi| ×∑

j αj|θj|. One can easily check that

Dg·W =

gj1...

gj1...

gjt...

gjt

DW

g−1i1

...g−1i1

...g−1is

...g−1is

So if DW is a square matrix the determinant of DW is a θ-semi-invariant:

detDg·W = det gθj1j1· det g

θjtjt

detDW det g−|θi1 |i1

· det g−|θis |is

= gθ detDW .

We will call these semi-invariants determinantal semi-invariants

Theorem 5.2. As a C[RepαQ]GLα-module C[RepαQ]θ is generated by determinan-tal semi-invariants. As a ring SIθ[RepαQ] is generated by invariants (i.e. tracesof cycles) and determinantal nθ-semi-invariants with n ∈ N.

Note that this implies that there are only θ-semi-invariants if DW is a squarematrix so

∑i αi|θi| =

∑j αj|θj| or equivalently θ · α = 0.

Now we can use this special form for the semi-invariants to get a nice interpreta-tion for the covering of Rep(Q,α)ssθ//GLα.

As we have seen Rep(Q,α) describes the α-dimensional representations of thepathalgebra CQ. Now if W ∈ Rep(Q,α) is θ-semistable then there exists a∑

j |θj| ×∑

i |θi|-matrix D with entries in CQ such that detDW 6= 0, so DW isan invertible matrix:

∃EW : DWEW = 1∑ |θj |αj and DWEW = 1∑ |θi|αi23


So W is also a representation of a new algebra for which D is is indeed aninvertible matrix. To be more precise we need a good interpretation a the identitymatrices that appeared in the equations above. Recall that for an α-dimensionalrepresentation W the vertex i can considered as an idempotent in CQ and iW willcorrespond to the identity matrix on the αi-dimensional space iW . The identitymatrix 1∑ |θj |αj can hence be considered as the evaluation in W of the matrix

1j =

j1

...j1

...jt

...jt

=

[h(D11)

...h(Dpp)

]

Now we define E = Eµν to be the matrix such that DE = 1j and ED = 1i so∑κ

DµκEκν = δµνh(Dµµ) and∑κ

EµκDκν = δµνt(Dµµ).

Now we define the universal localization of CQ at D to be the algebra

CQ[D−1] = CQE/(∑κ

DµκEκν − δµνh(Dµµ),∑κ

EµκDκν = δµνt(Dµµ))

Here QE is a new quiver consisting of Q together with extra arrows Eµν such thath(Eµν) = t(Dµν) and t(Eµν) = h(Dµν).

There is a natural map CQ→ CQ[D−1] so we also have a map

Rep(CQ[D−1], α)→ Rep(Q,α)

This map is an (open) embedding because (D−1)W is uniquely defined by DW ,its image consist precisely of these representations of CQ for which detDW 6= 0.

Theorem 5.3. Repssθ (Q,α) can be covered by representation spaces of universallocalizations of CQ. This covering is compatible with the GLα-action, so Mss

θ (Q,α)can be covered by quotient spaces of universal localizations of CQ.

This theorem also holds for quotients of path algebras. We will work this out inthe next section for the preprojective algebras.

5.3 Moduli space for the smash product

So lets now take a closer look at the case of the singularity C[X, Y, Z]/(XY −Zn).As we already know we can consider the singularity as the quotient space of the

24


preprojective algebra over the McKay quiver with the standard dimension vector.

V//G = iss(ΠG, αG.

In order to construct a nice desingularization of this space we have to find a goodcharacter. Let e1, . . . , en be the vertices of the quiver and let e1 correspond to thetrivial representation. For every vertex ei 6= e1 there exists a character θi mappinge1 to −αGi = − dimSi, ei to 1 and all the other vertices to zero. We denote thesum of all these θi as θ and this will be the character under consideration:

θ(e1) = −|αG|+ 1 and θi = 1 if i 6= 0.

Theorem 5.4. If V//G is a kleinian singularity and V //G→ V//G is its minimalresolution, then V //G = Mss

θ (ΠG, α).

The proof of this theorem can be found in [?]. We will just show how one cansee the equivalence in the A case.

The semi-invariants are constructed using matrices D of which the entries are allpaths starting from e1. If we construct DW , then every column of DW correspondsto a column of D because the dimension of e1 is 1. The determinant is linearin the columns so we can chose D up to linear combinations of the columns. Inthis way we can turn D into a form such that DW is block diagonal with thedimension of every block corresponding to the dimension of a vertex. This meansthat the θ-semi-invariants are generated by products of the θi-semi-invariants.Therefore we can conclude that a representation is θ-semistable if and only if itis θi-semistable for every i.

The θi-semi-invariants are generated over C[iss(Π, α)] by D’s that are 1×dimSi-matrices whose entries are paths from e1 to ei. Using the preprojective relationand the relations from matrix identities one can find a finite number of gener-ating paths. For instance, these paths cannot run twice through a vertex withdimension 1 otherwise we could split of a trace of a cycle (and this is containedin C[iss(Π, α)]). If there are ki such paths there are Cki

αigenerators for the semi-

invariants.

This means that we can embed M ssθ (Π, α) in

C3 × PCk2α2 × · · · × PC

knαn .

The first factor is for the 3 invariants, the others for the θi-semi-invariants forevery i > 1.

In the A-case, up to multiplication with invariants there are for every ei exactlytwo paths from e1: a clockwise pi and a counterclockwise qi. Each of these paths

25


gives a θi-semi-invariant. The projection map Repssθ (Π, α)→M ssθ (Π, α) can now

be seen asW 7→ [(XW , YW , ZW ), (p2, q2)W , . . . , (pn, qn)W ].

To calculate the exceptional fiber we must look at the semistable representationthat have zero invariants (X, Y, Z). Because X is zero there must be an i suchthat piW 6= 0 but pi+1W = 0. Semistability then implies that qi+1W 6= 0. Alsoqi−1W must be zero otherwise Z = pi/pi−1qi−1/qi would not be zero. This meansthat a point P comes from a point in the exceptional fiber if it is of the form

[(0, 0, 0), (1, 0), · · · , (1, 0), (pi, qi)W , (0, 1), · · · , (0, 1)].

From this we can conclude that the exceptional fiber is indeed the union of n− 1P1 intersection each other consecutively.

26

Task

• Choose your favorite finite abelian group of at least 3 elements.

• Construct a linear action of this group on either C2 of C3 that is faithfull(i.e. no element of the group acts trivially).

• Calculate the generators and relations of the ring of invariants RG for thisaction.

• Find a presentation of the smash product as a path algebra of a quiver withrelations.

• Describe the representation space for dimension vector (1, . . . , 1) and checkthat its quotient corresponds indeed the ring of invariants. Describe theorbits of this space.

• Describe the moduli space for dimension vector (1, . . . , 1) and character(−n+ 1, 1, . . . , 1)

• Describe the exceptional fiber over the zero. itemize

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